1. ⇒ (MHT CET 2023 10th May Morning Shift )

The points $(1, 3), (5, 1)$ are opposite vertices of a diagonal of a rectangle. If the other two vertices lie on the line $y = 2 x + c$ , then one of the vertex on the other diagonal is

A. $(1, - 2)$

B. $(0, - 4)$

C. $(2, 0)$

D. $(3, 2)$

Correct answer option is (C)

MHT CET 2023 10th May Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 7 English Explanation

Diagonals of rectangle bisect each other.

$∴$ Midpoint of $(1, 3)$ and $(5, 1)$ is $(3, 2)$ .

Also, $y = 2 x + c$ passes through $(3, 2)$ .

$\begin{array}{ll} ∴ 2 & = 2 (3) + c \\ ∴ & c = - 4 \end{array}$

$∴$ Other two vertices lie on $y = 2 x - 4$ .

Let co-ordinates of $B$ be $(x, y)$ i.e., $(x, 2 x - 4)$

slope of $AB \times$ slope of $BC = - 1$

$\begin{aligned} \Rightarrow (\frac{2 x - 4 - 3}{x - 1}) (\frac{2 x - 4 - 1}{x - 5}) = - 1 \\ \Rightarrow (\frac{2 x - 7}{x - 1}) (\frac{2 x - 5}{x - 5}) = - 1 \\ \Rightarrow x^{2} - 6 x + 8 = 0 \\ \Rightarrow x = 4, 2 \end{aligned}$

When $x = 4, y = 4$

When $x = 2, y = 0$

$∴$ Vertex of the other diagonal is $(2, 0)$ .

2. ⇒ (MHT CET 2023 9th May Morning Shift )

The distance of a point $(2, 5)$ from the line $3 x + y + 4 = 0$ measured along the line $L_{1}$ and $L_{1}$ are same. If slope of line $L_{1}$ is $\frac{3}{4}$ , then slope of the line $L_{2}$ is

A. $- \frac{3}{4}$

B. $\frac{1}{3}$

C. $\frac{1}{4}$

D. 0

Correct answer option is (D)

MHT CET 2023 9th May Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 15 English Explanation

According to the given condition, in $△ ABC, AB = AC$ .

$∴ △ ABC$ is an isosceles triangle.

Let $m, m_{1}, m_{2}$ be the slopes of given line, $L_{1}$ and $L_{2}$ respectively.

$\begin{array}{ll} ∴ & m = - 3, m_{1} = \frac{3}{4} \\ ∴ & | \frac{m - m_{1}}{1 + {mm}_{1}} | = | \frac{m - m_{2}}{1 + {mm}_{2}} | \\ ∴ & 3 = | \frac{- 3 - m_{2}}{1 + 3 m_{2}} | \\ \Rightarrow m_{2} = 0 \end{array}$

3. ⇒ (MHT CET 2021 20th September Morning Shift )

The slope of the line through the origin which makes an angle of 30 $^{\circ}$ with the positive direction of Y-axis measured anticlockwise is :

A. $\frac{- 2}{\sqrt{3}}$

B. $- \sqrt{3}$

C. $\frac{\sqrt{3}}{2}$

D. $\frac{- 1}{\sqrt{3}}$

Correct answer option is (B)

MHT CET 2021 20th September Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 27 English Explanation

Refer figure.

Angle made by line L with positive direction of X axis is (90 $^{\circ}$ + 30 $^{\circ}$ ) i.e. 120 $^{\circ}$ .

$∴$ Slope of line L = $\tan (120^{\circ}) = \tan (π - 60^{\circ}) = - \tan 60^{\circ} = - \sqrt{3}$

⇒Straight Line